Topological trapping in circular midpoint opinion dynamics

Abstract

We study a discrete-time asynchronous midpoint dynamics on the circle in which, at each step, a uniformly chosen neighboring pair moves to the midpoint along the shortest arc. Although the update rule is locally contractive, we show that the global relaxation mechanism depends sharply on the boundary topology. Under open boundary conditions the system converges almost surely to consensus through pure contraction. Under periodic boundary conditions the graph contains a single cycle, and the wrapped edge increments define an integer-valued winding number. While consensus remains the unique absorbing state for every fixed system size, we show that topology profoundly reshapes the transient dynamics. We prove that branch-crossings are the only mechanism capable of modifying the winding number and compute explicitly their probability for disordered initial data. Local averaging rapidly suppresses large gradients and drives the system into a no-branch-crossing regime where the winding number freezes. Inside a fixed winding sector we construct an adaptive co-moving frame in which the dynamics becomes an exact Euclidean midpoint process and establish strict contraction toward a twisted linear profile determined by the winding number. Our results isolate a minimal mechanism by which a single cycle induces sector locking and escape, even though the final equilibrium remains unchanged.

Figures (17)

• Figure 1: Midpoint update under the ACCA dynamics. (a) If $|\operatorname{wrap}_\pi(\theta_t(k+1)-\theta_t(k))|<\pi$, the shortest arc is unique and both angles move to its midpoint. (b) If $|\operatorname{wrap}_\pi(\theta_t(k+1)-\theta_t(k))|=\pi$, the two angles are antipodal and there are two shortest semicircles; the update chooses one midpoint uniformly with probability $1/2$.
• Figure 2: Example illustrating that the midpoint along the shortest arc does not coincide with the wrapped arithmetic mean. Here $\theta_t(k)=0.9\pi$ and $\theta_t(k+1)=-0.9\pi$. The update yields $-\pi$, while the wrapped arithmetic mean equals $0$.
• Figure 3: Simulation for a system with $N=200$ with empty boundary conditions (a) is the starting opinion distribution (b) configuration at $t=2.5\cdot 10^7$ with temporary ladder structures (c) at $t=5\cdot 10^7$ only one global ladder is remaining (d) at $t=10^{8}$ the ladder is reduced
• Figure 4: Simulation for a system with $N=200$ with empty boundary condition as in Figure \ref{['fig:200_empty']} in cylindrical coordinates
• Figure 5: Simulation for a system with $N=1000$ with empty boundary conditions (a) is the starting opinion distribution (b) configuration at $t=2.5\cdot 10^9$ with temporary ladder structures (c) at $t=5\cdot 10^9$ only one global ladder is remaining (d) at $t=10^{10}$ the ladder is reduced

Theorems & Definitions (42)

• Definition 2.1: Total increment and winding number
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Lemma 3.1: Consensus with open boundary conditions
• proof : Proof sketch
• Proposition 3.2: Consensus with periodic boundary conditions
• proof : Proof sketch
• Theorem 4.1: Branch-crossing and winding number
• proof